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Hamming weights and Betti numbers of Stanley–Reisner rings associated to matroids

Abstract

To each linear code \(C\) over a finite field we associate the matroid \(M(C)\) of its parity check matrix. For any matroid \(M\) one can define its generalized Hamming weights, and if a matroid is associated to such a parity check matrix, and thus of type \(M(C)\), these weights are the same as those of the code \(C\). In our main result we show how the weights \(d_1,\ldots ,d_k\) of a matroid \(M\) are determined by the \(\mathbb N \)-graded Betti numbers of the Stanley–Reisner ring of the simplicial complex whose faces are the independent sets of \(M\), and derive some consequences. We also give examples which give negative results concerning other types of (global) Betti numbers, and using other examples we show that the generalized Hamming weights do not in general determine the \(\mathbb N \)-graded Betti numbers of the Stanley–Reisner ring. The negative examples all come from matroids of type \(M(C)\).

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References

  1. 1.

    Bayer, D., Stillman, M.: Computation of Hilbert functions. J. Symb. Comput. 14, 31–50 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Bigatti, A.: Upper bounds for the Betti numbers of a given Hilbert function. Comm. Algebra 21(7), 2317–2334 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Björner, A.: The Homology and Shellability of Matroids and Geometric Lattices. Matroid Applications, Encyclopedia Math. Appl., vol. 40, pp. 226–283. Cambridge University Press, Cambridge (1992)

  4. 4.

    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Britz, T., Johnsen, T., Mayhew, D., Shiromoto, K.: Wei-type duality theorems for matroids. Des. Codes Cryptogr. 62(3), 331–341 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Bruns, W., Hibi, T.: Stanley–Reisner rings with pure resolutions. Comm. Algebra 23(4), 1201–1217 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Duursma, I.M.: Combinatorics of the Two-Variable Zeta Function, Finite Fields and Applications, pp. 109–136, Lecture Notes in Comput. Sci., 2948. Springer, Berlin (2004)

  8. 8.

    Eagon, J.A., Reiner, V.: Resolutions of Stanley–Reisner rings and Alexander duality. J. Pure Appl. Algebra 130(3), 265–275 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Hochster, M.: Cohen–Macaulay Rings, Combinatorics, and Simplicial Complexes. Ring Theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pp. 171–223. Lecture Notes in Pure and Appl. Math., vol. 26. Dekker, New York (1977)

  10. 10.

    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity and Computer Computations, pp. 85–103. Plenum Press, New York (1992)

  11. 11.

    Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)

  12. 12.

    Oxley, J.: Matroid Theory. Second Edition. Oxford Graduate Texts in Mathematics, vol. 21. Oxford University Press, Oxford (2011)

    Google Scholar 

  13. 13.

    Roune, B.: A slice algorithm for corners and Hilbert-Poincaré series of monomial ideals. In: ISSAC 2010, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, pp. 115–122. ACM, New York (2010)

  14. 14.

    Sáenz-de-Cabezón, E.: Multigraded Betti numbers without computing minimal free resolutions. Appl. Algebra Eng. Comm. Comput. 20(5–6), 481–495 (2009)

    MATH  Article  Google Scholar 

  15. 15.

    Sáenz-de-Cabezón, E., Wynn, H.P.: Betti numbers and minimal free resolutions for multi-state system reliability bounds. J. Symb. Comp. 44, 1311–1325 (2009)

    MATH  Article  Google Scholar 

  16. 16.

    Stanley, R.P.: Combinatorics and Commutative Algebra. Second edition. Progress in Mathematics, vol. 41. Birkhäuser, Boston (1996)

    Google Scholar 

  17. 17.

    Vardy, A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inform. Theory 43(6), 1757–1766 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Wei, V.K.: Generalized Hamming weights for linear codes. IEEE Trans. Inform. Theory 37(5), 1412–1418 (1991)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgments

Trygve Johnsen is grateful to Institut Mittag-Leffler where he stayed while a part of this work was being done. The authors are grateful to the referees and editor for their suggestions for improving the original paper.

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Correspondence to Hugues Verdure.

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Johnsen, T., Verdure, H. Hamming weights and Betti numbers of Stanley–Reisner rings associated to matroids. AAECC 24, 73–93 (2013). https://doi.org/10.1007/s00200-012-0183-7

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Keywords

  • Codes
  • Matroids
  • Stanley–Reisner rings

Mathematics Subject Classification (2000)

  • 05E45
  • 94B05
  • 05B35
  • 13F55